How would you take math notes to make the most of them?

7Viliam

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To me it helps to imagine that I am *explaining the topic to someone else*. If I had enough time, I would never *copy *the textbook; I would rewrite it using my own words, and probably change the entire structure. (In other words, instead of "paper1 -> paper2", it would go "paper1 -> internal model -> paper2".) Unfortunately, doing things the way I wish takes *a lot *of time.

For example, if I make notes about programming, I am trying to write the simplest code that illustrates the concept *in isolation from other concepts*. (Most examples I find online are introducing multiple concepts at the same time. Okay, I suppose in reality, you usually use X and Y and Z together in the same project. But I still want to see X used separately, and Y separately, and Z separately. And *then *an example of how X and Y and Z go together.)

I would suggest to explore the concept in unusual ways. For example, when you learn about commutative operators, don't just use "addition" and "multiplication" as obvious examples, but also think about ones like "least common multiple" or even "these words have the same amount of strokes in Chinese". (Ultimately coming to "there is an arbitrary undirected graph, where the nodes are the possible inputs, and each edge contains an arbitrary output as a label".)

Also, when you learn things, the value is not merely in the individual things, but also (mostly?) in their *connections *to other things. That is the difference between a newbie who can recite the facts but cannot apply them, and an expert who can immediately take three abstract concepts and chain them together to solve a problem. (Not sure what exactly this imples for note-taking and zettelkasten method. My preferred way to make notes would be like making wiki pages, so I would mention these connections at the bottom of the page.) For example, there are many proofs that there are infinitely many primes, but I enjoyed reading an argument how having finitely many primes would allow us to create an insane compression algorithm. (You take the input as a binary number, factorize it, and save the factors. If your input is much larger than the hypothetical largest prime, the output file size will be a logarithm of the input file size.)

It might depend on the area of math.

Things I have found useful:

1. Make up problems, and solve them. (Calculus: what is the volume of a sphere? Not using 'the' formula - integrate it from top to bottom. (Getting to the general formula eventually requires plugging in variables or functions rather than numbers.) There's also a^b=b^a.)

2. Find different 'languages' and convert between them. (It may seem obvious how to convert between addition and multiplication: 5+5+5 = 3x5 = 3+3+3+3+3. But what is 1+2+3+...+100?)

3. Generalizing (The easiest explanations are 'make functions/algorithms'. 3+5=5+3=8 replaced by f(3, 5) = 8, perhaps because f(3, 5) = f(2, 6) = f(1, 7) = f(0, 8) = 8. Less trivially: Everything is a function. Derivative operator? Turns functions into (usually other) functions. (Looking into what functions it doesn't change/are their own derivatives might be useful for reasons similar to why knowing about 1 is good for understanding/working with multiplication.))

4. What makes sense to you? Some people find it easier to understand derivatives by analogy to the (related) discrete case: If one of your character's stats in a video game automatically go up by some amount when your character levels up, then the amount a stat goes up between level 11 and 12 is like the derivative of that stat with regards to level.* Aside from the particular mathematical connections between these two things, the advantage is that something has been put in other terms: discrete rather than continuous.

5. Just practicing might be more important than note taking. (If I want to get good at trig functions, I don't have amazing notes on trig functions - I have a set of starting equations ("identities") which I can take and combine and fiddle with, and then I will be better with handling trig functions.) I guess I'm saying "you don't have to work with "first" principles, but working from a starting point gets you somewhere."

Note that 4 ties into 2.

*There's a post here on LW about that, though it didn't dive into *why* that was a great connection.

I've been heavily influenced by reading about the zettelkasten method (and its siblings); it seems to, theoretically, solve many problems that I (a bioinformatics student) encounter on a daily basis.

However, I'm struggling with implementing said method in practice. While in biology I think the it will be easily applicable, in math I have some trouble with both splitting the information into chunks (zettels, if you wish)

andthen with linking those chunks together in a meaningful and useful way.As an illustrative example, let's pick analysis, with the common

theorem-proof-definition-proposition-proofstructure (think Baby Rudin). How would you split the information, and how would you link it together?Another problem I'm having concerns note-taking in general; especially in math I often feel I'm just copying the textbook, and that there's no added value in my notes. What information should go in my notes, as opposed to just staying in the textbook?